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The existence of time-dependent attractor for wave equation with fractional damping and lower regular forcing term

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Luo is supported by NSF grant(11961059) and "Innovation Star" of Gansu Provincial Department of Education (2021CXZX-206)

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  • We investigate the well-posedness and longtime dynamics of fractional damping wave equation whose coefficient $ \varepsilon $ depends explicitly on time. First of all, when $ 1\leq p\leq p^{\ast\ast} = \frac{N+2}{N-2}\; (N\geq3) $, we obtain existence of solution for the fractional damping wave equation with time-dependent decay coefficient in $ H_{0}^{1}(\Omega)\times L^{2}(\Omega) $. Furthermore, when $ 1\leq p<p^{*} = \frac{N+4\alpha}{N-2} $, $ u_{t} $ is proved to be of higher regularity in $ H^{1-\alpha}\; (t>\tau) $ and show that the solution is quasi-stable in weaker space $ H^{1-\alpha}\times H^{-\alpha} $. Finally, we get the existence and regularity of time-dependent attractor.

    Mathematics Subject Classification: Primary: 35B41, 37L30; Secondary: 35L05.


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